EDB — 0GQ

[0GQ] Prerequisites:[06N]↺↻,[06M]↺↻,[06V]↺↻.Difficulty:*.(Replaces 29W)

Let \((X,\tau )\) be a topological space. Consider the descending ordering between sets  ¹ , with this ordering \(𝜏\) is a directed set; we note that it has minimum, given by \(∅\).

Now suppose the topology is Hausdorff. Then taken \(x∈ A\), let \({{\mathcal U}}=\{ A∈𝜏: x∈ A\} \) be the family of the open sets that contain \(x\): show that \({{\mathcal U}}\) is a directed set; show that it has minimum if and only if the singleton \(\{ x\} \) is open (and in this case the minimum is \(\{ x\} \)). ²

By the exercise [06V]↺↻, when \(\{ x\} \) is not open then \({{\mathcal U}}\) is a filtering set, and therefore can be used as a family of indices to define a nontrivial ”limit” (see Remark [237]↺↻). We will see applications in section [2B8]↺↻.